Normal Distribution

The normal distribution is

f(x) = (¹⁄_{σ √2π}) e^{-(x-µ) ²⁄_2σ ²}

Normal distribution

Probability density function Probability density function for the normal distribution The red curve is the standard normal distribution
Cumulative distribution function Cumulative distribution function for the normal distribution
Notation	${\mathcal {N}}(\mu ,\,\sigma ^{2})$
Parameters	μ ∈ R — mean (location) σ2 > 0 — variance (squared scale)
Support	x ∈ R
PDF	${\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}$
CDF	${\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]$
Mean	μ
Median	μ
Mode	μ
Variance	$\sigma ^{2}\,$
Skewness	0
Kurtosis	0
Entropy	${\frac {1}{2}}\ln(2\pi e\,\sigma ^{2})$
MGF	$\exp\{\mu t+{\frac {1}{2}}\sigma ^{2}t^{2}\}$
CF	$\exp\{i\mu t-{\frac {1}{2}}\sigma ^{2}t^{2}\}$
Fisher information	${\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}$

Curve Equation

CDF

PDF

Half Normal Distribution Equations

• The parameter μ in this formula is the mean or expectation of the distribution (and also its median and mode).
• The parameter σ is its standard deviation; its variance is therefore σ'². A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.